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Continuing the discussion from Neither a cessationalist, nor a permanent consciousness:
Thread to discuss LEM, double negation elimination, direct vs indirect proofs, constructive logic, intuitionism, the title, and where @Pj_Harivarmanian disagrees with me 
PS @dogen shout out
Hi, @yeshe.tenley , thanks for starting the new thread. I actually asked you to summarize your position in this one, though. The original thread is pretty longācould you summarize your point here, please?
In the previous thread, I responded to you:
by agreeing. Unfortunately. due to Aristotleās error, most assume LEM is required as a law of thought. However, on this forum youāll find some who do not make this mistake. I then gestured to a lot of old posts and threads on this forum.
I also made the claim that the Teacher:
takes a decidedly constructivist approach.
To which you disagreed?
To be fair to you, what I said does have the flair of being anachronistic. Let me try something a bit more conservativeā¦
I do not believe the Teacher ever made use of double negation elimination to prove his points logically. It is in that sense I made the above anachronistic claim. Where the Teacher employed logic to prove a point I do not believe he ever made use of double negation elimination nor do I think he would have considered someone who did to have employed a valid logical operation.
Whatever logical system the Teacher used, I do not think it included double negation elimination as a valid logical step. I wonder if you disagree with this or can find some counterpoint example.
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I opened this thread because you seem educated on this topic and because you said this:
Yes, I read your forum posts about LEM a while agoāgood stuff, really. I disagree with quite a bit
I simply would like to know what you disagree with in the hopes I might learn something. 
I agree itās anachronistic, but letās overlook that for now just to advance the argument
Iām really not qualified to judge if historical texts support this one way or another ā there are too many alternative readings. But for the sake of argument, letās assume they do, just to isolate where I disagree.
1.1. Under that assumption: double negation is only an issue if you lack a proof for it.
The difference between classical logic (like Frege, bypassing Aristotle for other reasons) and Heytingās intuitionist logic is that classical eliminates double negation (neg-neg-P to P) axiomatically, whereas Heyting requires an explicit proof to assert neg-neg-P ā P.
Personally, I think very few empirical problems actually fit this distinction, and I highly doubt the historical texts operated with that level of logical precision. There was no need for that, I think.
3.1. That said, what you must logically demonstrate is that the specific instances where you claim the Teacher denied the law of double negation cannot be interpreted in any other way.
3.1.1. Even if I grant you that he denied it, my question is: did he deny it because he lacked a proof that neg-neg-P leads to P, or because he possessed a proof that neg-neg-P does not lead to P?
3.1.1.1. Whichever you claim, you must provide the semantic content of that position within that specific context. Here is an example of what I mean by semantic content:
3.1.1.1.1. Intuitionists accept that not-not-Even(2) implies Even(2) because you can prove double negation holds for this exact case. The intuitionist denial of the excluded middle or double negation isnāt an axiomatic rule like in classical logicāit would be a problem if it were.
3.1.1.2. Intuitionist logic is intuitionist precisely because it doesnāt blindly accept these laws as axioms. But that doesnāt mean it rejects them completely. On the contrary, it accepts them whenever they can be proved.
I hope youāll notice that I disagree with you on quite a few points, though perhaps not exactly in the ways you expected. Either way, thereās a lot to unpack in what Iāve written above, but I think youāll particularly like point 3.1.1.
If anything isnāt clear, just let me know and Iāll try to clarify.
Thanks, @yeshe.tenley .
Donāt you think that Tetralemma being the valid options makes it seem the logic that the PÄli Canon is operating with doesnāt hold double negation elimination (DNE) axiomatically?
ā If the Canon held DNE axiomatically, āNeither X, Nor Non-Xā would never be an option to begin with.
ā āNeither X, Nor Non-Xā is an option
ā Therefore, it seems DNE isnāt held axiomatically.
I think āUniverse is neither finite, nor infiniteā fits the bill (and if itās due to category error - what category is there in assuming universe?)
OTOH, for a āBoth X and Not-Xā, thereās the sutta SN 41.7:
āHouseholder, the limitless release of the heart, and the release of the heart through nothingness, and the release of the heart through emptiness, and the signless release of the heart: do these things differ in both meaning and phrasing? Or do they mean the same thing, and differ only in the phrasing?ā
āSir, there is a way in which these things differ in both meaning and phrasing.
But thereās also a way in which they mean the same thing, and differ only in the phrasing."
So, it is both true that they mean the same thing, and they also differ in meaning.
The explanation in the sutta is as follows: Itās kind of like āTaking road A to arrive at Paradiseā vs āTaking road B to arrive at Paradiseā. Theyāre different roads, but in the end, theyāre the same destination.
In other words:
0A = 0B; differing in phrasing; differing in operation; same in results.
I think an example is from Iti 43:
There is, mendicants, freedom from rebirth, freedom from what has been produced, made, and conditioned. If there were no freedom from rebirth, freedom from what has been produced, made, and conditioned, then you would find no escape here from rebirth, from what has been produced, made, and conditioned. But since there is freedom from rebirth, freedom from what has been produced, made, and conditioned, an escape is found from rebirth, from what has been produced, made, and conditioned.
I know that in a previous thread Yeshe got into this sutta, but Iāll reframe it here as I understand it:
P = Rebirth
Q = Samsara
Ā¬P = Freedom from Rebirth
Ā¬Q = Escape from Rebirth/Samsara
So, this is probably the only DNE employed that Iāve seen in the suttas, and itās quite interesting that it uses it in a way compatible with intuitionist methods, without resorting to classical. 
Personally, that seems highly doubtful. In general, logicians reject logical laws using very specific quantifiers within narrow theoretical contextsāI can back this up with examples if needed. But thatās not how everyday empirical language works.
Axiomatically? Even worse.
In natural language, people use double denials all the time without explicitly spelling out what they mean. For example:
Does this imply they are rejecting the law of double negation like Heyting? No. Does that mean they are classical in the logic and foundations of math sense? No. In general, they know nothing about it.
We could easily find dozens of examples where ordinary speech doesnāt wear its logical form on its sleeve, leaving it open to many formal logical interpretations. Does that say anything extra about natural language? No.
Being compatible with intuitionist or constructivist methods is easyāplenty of things are. But that tells us almost nothing, because compatibility is a very low bar, especially for an intuitionist or any non-classical logician.
My point is that there is no way to settle anything solid about this by debating modern logic. What do I mean by that? I mean there really wonāt be any progress on what the Buddha meant by looking at it through this or that formal logic. Very few subjects can be better understood through that approach. Very few indeed. If someone wanders into that forest without a very good map, they could easily end up like Graham Priest, seeing actual contradictions in doors that were left wide open.
There are already established Theravadin, Sarvastivadin, and Sautrantika ways to read these passages without walking into this dark forest: injecting specialized formal logics does not add an ounce of real solidity to any of them. Formalising the water example to claim the speakers āwere intuitionistsā is like using quantum mechanics to explain a car accident.
Just out of curiosity,@Dogen : Iām a big fan of the late DÅgen from the 12-Fascicle ShÅbÅgenzÅ.
I only brought up category errors as an example. There are plenty of ways to read non-logical texts without dragging specialized logics into the mix.
Plus, whether something is a category error depends on what the opponent actually means by āfiniteā or āinfiniteā. Take a look at how the Sautrantikas and Theravadins interpret this passage and similar ones.
This is exactly backward. All I said is that the Teacher never himself used double negation elimination to prove an assertion. He did not employ this tool as part of the proof machinery.
All you need do to throw some doubt on this is find some plausible places where he did. I have looked and cannot find any. You are welcome to try and I would be very interested if you were able to come up with some.
Double negation elimination is not a sophisticated logical procedure. Common folks - non logicians - use it all the time to varying degrees of success. It isnāt like weāre talking about some rarified logical abstraction - like the Curry-Howard Isomorphism hehe - here.
Iām saying he did not employ it to prove his points. At least not that I can find. See if you can find some plausible evidence to the contrary in sutta!
Sure! But it does not employ them as part of the proof machinery. Indirect proofs are not allowed. Find a place the Teacher plausibly employed double negative elimination as axiomatic and Iāll be interested.
Please give a logical system where the tetralemma is axiomatic: the interlocutors discussing it grant that all four corners can be inhabited, but that also allows double negation elimination as an axiomatic procedure.
Classical logic is the specialized one here. It allows for double negation elimination axiomatically. The logics you get by not assuming this are less specialized than so-called classical.
I think you did not understood me. Apologies, I think I was not clear.
Let me try another approach. I will simply concede that there is no instance where the Buddha explicitly rejected the law of double negation. As I mentioned, finding or not finding such examples depends on a specialized, rigorous exegesis of the Suttas, which is beyond my scope. To clarify my previous statement: I am conceding this premise because it ultimately does not affect the point I am trying to make.
That is, the intuitionistic way to frame the question remains: when that rule is absent, did the Buddha lack a proof for either : āPā or āneg-neg-Pā? Or did he only had a proof for āneg-neg-Pā but not for āneg-neg-P ā Pā? ? What do you think is the case here?
Constructive or intuitionistic logics are just as specialized as classical logic. Why wouldnāt they be? I donāt quite get what you mean by that, but either way, that part doesnāt matter to my point.
I disagree. But this is an empirical matter; we cannot resolve it through thought. The proper approach requires statistical and linguistic empirical research. Since that is also irrelevant to my point, so I will concede that as well.
If he had a proof of neg-neg-P, then he didnāt automatically travel over the bridge and accept P as proven. He might have arrived at P by some other means, but not by the general bridge of Ā¬Ā¬P->P.
If he wanted to prove P, then he would have required a direct witness of P and not just accepted Ā¬Ā¬P as that direct witness.
To be very clear, he might even accept a direct witness that Ā¬Ā¬P->P for some particular P, but he would not accept that holds for all P.
Because if you take constructive logic and add the general bridge of Ā¬Ā¬P->P you get classical. So in that sense classical is more specialized. It adds something. Just donāt travel over that Ā¬Ā¬P->P bridge and you are in constructive land.
I can concede that. If he arrives at P by other means, then he accepts the logical law of bivalence, but not the law of double negation?
Yes, agreed, but I do not understand what this has to do with constructive or intuitionistic logics being specialised. Let me clarify what I mean by āspecialisedā: I mean specialised as a discipline of human knowledge. It is not common, ordinary knowledge; it is a field that requires formal training and study.